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Low Power Warp Drive for Dummies jacksarfatti@icloud.com March 27, 2019 ε11 ε12 ε13 ε ij ≡ ε 21 ε 22 ε 23 ε 31 ε 32 ε 33 3D spacelike electric permittivity tensor for spatially inhomogenous-time dependent anisotropic matter made out of electrically charged particles (charge neutral over-all) in an inertial rest frame for the material In 4D special relativity in the same local inertial rest frame ε 00 ε 01 ε 02 ε 03 ε IJ = ε10 ε11 ε12 ε13 ε 20 ε 21 ε 22 ε 23 ε 30 ε 31 ε 32 ε 33 Similarly, for the magnetic permeability tensor. Their local frame invariant inner product is ε IJ µ IJ = ε 00 µ 00 + ε 01µ 01 + ε 02 µ 02 + ε 03µ 03 + ε10 µ10 + ε11µ11 + ε12 µ12 + ε13µ13 + ε 20 µ 20 + ε 21µ 21 + ε 22 µ 22 + ε 23µ + ε 30 µ 30 + ε 31µ 31 + ε 32 µ 32 + ε 33µ 33 The same material seen by a locally coincident observer in a properly tensorial accelerating (non-inertial) local frame is given by the tetrad transformation1 1 Similarly for Lorentz transformations between locally coincident inertial frames given by Λ II ' . ε µν = ξ µI ξνJ ε IJ = ξ µ0ξν0ε 00 + ξ µ0ξν1ε 01 + ξ µ0ξν2ε 02 + ξ µ0ξν3ε 03 ξ µ1 ξν0ε10 + ξ µ1 ξν1ε11 + ξ µ1 ξν2ε12 + ξ µ1 ξν3ε13 ξ µ2ξν0ε 20 + ξ µ2ξν1ε 21 + ξ µ2ξν2ε 22 + ξ µ2ξν3ε 23 ξ µ3ξν0ε 30 + ξ µ3ξν1ε 31 + ξ µ32ξν ε 32 + ξ µ3ξν3ε 33 However, both the local tetrad and the local Lorentz transformation are orthogonal therefore, the inner tensor product of the electric permittivity and magnetic permeability tensor are absolute locally objectively real invariants at the given physically real “local coincidence” in the sense of Einstein’s “hole problem.”2 Therefore, the actual speed of light in matter including both virtual vacuum electrical charges and real electrical charges comprising matter (overall charge neutrality as in a plasma) is an absolute local observable frame invariant for all physically possible local observers. That is, ε IJ µ IJ = ε I ' J ' µ I ' J ' = ε µν µ µν = ε µ 'ν ' µ µ 'ν ' Where I → I ' is a local Lorentz transformation between inertial frames I → µ is a local Tetrad transformation between an inertial and a non-inertial frame µ → µ ' is a local general coordinate transformation between non-inertial frames. Maxwell’s unification of electricity, magnetism and light is his constituent equation c= 2 1 ε µν µ µν Carlo Rovelli’s “Quantum Gravity” Chapter 2 for details on how to picture the physical meaning of general coordinate transformations in absolute terms in which naïve Cartesian coordinates have no physical meaning. Lee Smolin calls this “background independence.” ( Gµν = 8π G ε γδ µ γδ 2 )T µν Dielectric permittivity 4D tensor splits into "Vacuum" + "Matter" ZPF (Zero Point Fluctuations, off-mass-shell) Matter (real particles on-mass-shell) ε γδ ≡ ε γδ ( ZPF virtual particles ) + ε γδ ( real particles ) Similarly for magnetic permeability. ε γδ µ γδ = ε γδ ( ZPF ) µ γδ ( ZPF ) + ε γδ ( ZPF ) µ γδ ( Matter ) + ε γδ ( Matter ) µ γδ ( ZPF ) + ε γδ ( Matter ) µ γδ ( Matter ) ε γδ µ γδ → ∞ ⇒ Tµν → 0 for fixed Gµν